FACES, EDGES, AND VERTICES, OH MY! WHY? Author: Peggy Harvey Subject: Math Topic: Geometric shapes Grade Level: 7 UNIT DESCRIPTION In this geometry investigation, students explore the properties of geometric solids, using nets and online interactive sites, then use the relationship between these properties to develop Euler’s formula. LESSON DESCRIPTION FOR Day Students will first review polygons and circles using cardstock cut-outs. Using the same cut-outs, students will make comparisons between ways polygons can be assembled into 3-dimensional shapes called geometric solids. Students must decide how to build them on their own. Using nets, students will construct geometric solids (prisms, pyramids, cylinders, cones) and record their characteristics(faces, edges and vertices) in a chart. They will use an interactive website to check their findings. Finally, they are asked to look for a relationship between the number of faces, edges and vertices, and develop Euler’s Formula. TEKS (7th grade) (5) Patterns, relationships, and algebraic thinking. The student uses geometry and spatial reasoning, compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. The student is expected to: (C) use properties to classify three-dimensional figures, including pyramids, cones, prisms, and cylinders; (15) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to: (A) make conjectures from patterns or sets of examples and non examples; and (B) validate his/her conclusions using mathematical properties and relationships. LESSON GOALS The goals of this lesson are (1)to introduce the concept of geometric solids by building on students’ previous experiences with polygons and circles, (2)have students recognize the characteristics of geometric solids and (3)use those characteristics to develop Euler’s Formula (F + V = E + 2; Faces + Vertices = Edges + 2) METHODS NOTES: Be sure hearing-impaired student(s) are using their amplification device so they don’t miss any of the instructions or discussions. Have someone available to help sight0impaired student(s) explore using polygons and circles to form 3-dimensional shapes. ANTICIPATORY SET Have students use paper polygon and circle cut-outs to experiment with putting polygons and circles together to make 3-D shapes. (NOTE: print shapes and copy on cardstock, laminate, and cut out prior to lesson). Challenge gifted students by giving them a set number of shapes or specific shapes to use in their explorations. NOTE: Have someone help the sight-impaired studet(s) tape their shapes together. INTRODUCE AND MODEL KNOWLEDGE Using Smartboard technology, explore 3-D shapes and nets using online interactive activity at: http://www.learner.org/interactives/geometry/3d.html Also explore POLYHEDRA, PRISMS, and PYRAMIDS at same site. Have small plastic 3-D models of each shape for sight-impaired student(s) to explore during the Smartboard activity. GUIDED PRACTICE Each student should have a set of nets for the most common geometric solids and the Geometric Solids worksheet containing a chart with the names of the common geometric solids, and columns for recording the number of faces, vertices and edges of each solid. Even though students are building their own models, allow them to assist each other with the constructions. Students should discuss their findings and record them in the appropriate column of their worksheet chart. NOTE: Make one copy of the nets on cardstock and cut out for each sight-impaired student. Have them scored or pre-folded along each fold line. Also have someone available to help the sight-impaired student tape their nets into the 3-D shapes. Use interpretive software to load the chart onto the computer for sight-impaired students. Have a set of completed nets put back to use if slower students get frustrated and can’t figure out what shape a net should make. Use some of the less common nets for gifted students(octahedron, decahedron, dodecahedron, icosahedron, etc). INDEPENDENT PRACTICE Have students go to http://www.mathsnet.net/geometry/solid/index.html and test the entries on their chart for accuracy. Once all of the geometric solids have been constructed and the information recorded and tested, students should go to the interactive nets website http://www.learner.org/interactives/geometry/euler.html and enter their chart data. Their entries will be verified for accuracy and they will be guided through the process of using the entries in the chart to determine if the relationship between the number of faces, vertices and edges of 3-D solids. The site will guide them until they determine the relationship called Euler’s Formula. NOTE: Use website reading technology for sight-impaired students. WRAP-UP Discuss reasons someone would want to know Euler’s Formula, what occupations might use this formula, and where and how these geometric solids exist in the real world. ASSESSMENT Formative/ongoing assessment The worksheet provides a framework for the students to begin explorations and to record their findings. The interactive sites provide assessment throughout the activity. Summative/end of lesson assessment Students should complete the assessment activity at: http://www.learner.org/interactives/geometry/euler.html (Students should print their assessment results and turn in to teacher.) MATERIALS Materials Each group: Set of polygon shapes and circles cardstock cut-outs (laminated) 1 roll of scotch tape (at least 1 for every 2 students) graphing calculators(lower functioning students may need to use linear regression to fine formula) Copy of nets of common 3-D solids for each student:, rectangular prism, triangular prism, octagonal prism, square, pyramid, tetrahedron(less common ones for gifted students (octahedron, decahedron, dodecahedron, icosahedron, etc) Student Worksheet: with Geometric solids chart Computer access for each student Copy of nets on cardstock for sight-impaired students
Author: Peggy Harvey
Subject: Math
Topic: Geometric shapes
Grade Level: 7
UNIT DESCRIPTION
In this geometry investigation, students explore the properties of geometric solids, using nets and online interactive sites, then use the relationship between these properties to develop Euler’s formula.
LESSON DESCRIPTION FOR Day
Students will first review polygons and circles using cardstock cut-outs. Using the same cut-outs, students will make comparisons between ways polygons can be assembled into 3-dimensional shapes called geometric solids. Students must decide how to build them on their own. Using nets, students will construct geometric solids (prisms, pyramids, cylinders, cones) and record their characteristics(faces, edges and vertices) in a chart. They will use an interactive website to check their findings. Finally, they are asked to look for a relationship between the number of faces, edges and vertices, and develop Euler’s Formula.
TEKS (7th grade)
(5) Patterns, relationships, and algebraic thinking. The student uses geometry and spatial reasoning, compares and classifies two- and three-dimensional figures using geometric vocabulary and properties. The student is expected to: (C) use properties to classify three-dimensional figures, including pyramids, cones, prisms, and cylinders;
(15) Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to: (A) make conjectures from patterns or sets of examples and non examples; and (B) validate his/her conclusions using mathematical properties and relationships.
LESSON GOALS
The goals of this lesson are (1)to introduce the concept of geometric solids by building on students’ previous experiences with polygons and circles, (2)have students recognize the characteristics of geometric solids and (3)use those characteristics to develop Euler’s Formula (F + V = E + 2; Faces + Vertices = Edges + 2)
METHODS
NOTES: Be sure hearing-impaired student(s) are using their amplification device so they don’t miss any of the instructions or discussions. Have someone available to help sight0impaired student(s) explore using polygons and circles to form 3-dimensional shapes.
ANTICIPATORY SET
Have students use paper polygon and circle cut-outs to experiment with putting polygons and circles together to make 3-D shapes. (NOTE: print shapes and copy on cardstock, laminate, and cut out prior to lesson). Challenge gifted students by giving them a set number of shapes or specific shapes to use in their explorations.
NOTE: Have someone help the sight-impaired studet(s) tape their shapes together.
INTRODUCE AND MODEL KNOWLEDGE
Using Smartboard technology, explore 3-D shapes and nets using online interactive activity at:
http://www.learner.org/interactives/geometry/3d.html
Also explore POLYHEDRA, PRISMS, and PYRAMIDS at same site.
Have small plastic 3-D models of each shape for sight-impaired student(s) to explore during the Smartboard activity.
GUIDED PRACTICE
Each student should have a set of nets for the most common geometric solids and the Geometric Solids worksheet containing a chart with the names of the common geometric solids, and columns for recording the number of faces, vertices and edges of each solid. Even though students are building their own models, allow them to assist each other with the constructions. Students should discuss their findings and record them in the appropriate column of their worksheet chart.
NOTE: Make one copy of the nets on cardstock and cut out for each sight-impaired student. Have them scored or pre-folded along each fold line. Also have someone available to help the sight-impaired student tape their nets into the 3-D shapes. Use interpretive software to load the chart onto the computer for sight-impaired students. Have a set of completed nets put back to use if slower students get frustrated and can’t figure out what shape a net should make. Use some of the less common nets for gifted students(octahedron, decahedron, dodecahedron, icosahedron, etc).
INDEPENDENT PRACTICE
Have students go to http://www.mathsnet.net/geometry/solid/index.html and test the entries on their chart for accuracy. Once all of the geometric solids have been constructed and the information recorded and tested, students should go to the interactive nets website http://www.learner.org/interactives/geometry/euler.html and enter their chart data. Their entries will be verified for accuracy and they will be guided through the process of using the entries in the chart to determine if the relationship between the number of faces, vertices and edges of 3-D solids. The site will guide them until they determine the relationship called Euler’s Formula.
NOTE: Use website reading technology for sight-impaired students.
WRAP-UP
Discuss reasons someone would want to know Euler’s Formula, what occupations might use this formula, and where and how these geometric solids exist in the real world.
ASSESSMENT
Formative/ongoing assessment
The worksheet provides a framework for the students to begin explorations and to record their findings. The interactive sites provide assessment throughout the activity.
Summative/end of lesson assessment
Students should complete the assessment activity at:
http://www.learner.org/interactives/geometry/euler.html
(Students should print their assessment results and turn in to teacher.)
MATERIALS
Materials
Each group:
Set of polygon shapes and circles cardstock cut-outs (laminated)
1 roll of scotch tape (at least 1 for every 2 students)
graphing calculators(lower functioning students may need to use linear regression to fine formula)
Copy of nets of common 3-D solids for each student:, rectangular prism, triangular prism, octagonal prism, square, pyramid, tetrahedron(less common ones for gifted students (octahedron, decahedron, dodecahedron, icosahedron, etc)
Student Worksheet: with Geometric solids chart
Computer access for each student
Copy of nets on cardstock for sight-impaired students